We have developed a new crystal growth method named the traveling liquidus-zone (TLZ) method and found that the TLZ method is promising for growing compositionally homogeneous alloy crystals such as In₁₋ₓGaₓAs. Our one- dimensional TLZ growth model predicts precisely the sample translation rate for growing homogeneous In₁₋ₓGaₓAs crys- tals. In addition to the homogeneous crystal growth mode, supersaturation mode and supercooling mode exist in the TLZ method and these modes were examined experimentally and single crystals were found to be grown more easily when the supercooling controlled the growth process. Estimation of critical degree of supersaturation for avoiding poly- crystallinity is expected for further development of the TLZ method.